Prasad's comparison conjecture for second cohomology of p-adic analytic groups

From papers

Let PP be a pro-pp group, and write

Hcont2(P)=Hcont2(P,Fp),Hdisc2(P)=Hdisc2(P,Fp).H_{\operatorname{cont}}^{2}(P)=H_{\operatorname{cont}}^{2}(P,\mathbb{F}_p),\qquad H_{\operatorname{disc}}^{2}(P)=H_{\operatorname{disc}}^{2}(P,\mathbb{F}_p).

Let φ2:Hcont2(P)Hdisc2(P)\varphi^2:H_{\operatorname{cont}}^{2}(P)\to H_{\operatorname{disc}}^{2}(P) be the comparison map. Prasad's conjecture. For every pp-adic analytic group PP, the comparison map φ2\varphi^2 is an isomorphism.

This conjecture concerns when continuous and discrete cohomology agree for pro-pp groups. The comparison is known in particular for soluble pp-adic analytic and Chevalley pp-adic analytic groups, while the assertion for every pp-adic analytic group remains open based on the supplied context.

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Sources & referencesView supporting material

Primary source

Gustavo A. Fernandez-Alcober, Ilya V. Kazachkov, Vladimir N. Remeslennikov and Peter Symonds, “Comparison of the Discrete and Continuous Cohomology Groups of a Pro-p Group”, arXiv:math/0701737 (2007).

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